We show that a pair of chaotic systems subjected to the same noise may undergo a transition at large enough noise amplitude and follow almost identical trajectories with complete insensitivity to initial conditions. An analytic argument is presented to show that a pair of generic systems in the same potential evolving to equilibrium through standard Langevin dynamics with the same noise collapse into the same trajectory at long times.
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Maritan et al. (1994) studied this question.
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